4. Process Modeling
4.6. Case Studies in Process Modeling
4.6.3. Ultrasonic Reference Block Study

## Work This Example Yourself

View Dataplot Macro for this Case Study This page allows you to repeat the analysis outlined in the case study description on the previous page using Dataplot, if you have downloaded and installed it. Output from each analysis step below will be displayed in one or more of the Dataplot windows. The four main windows are the Output window, the Graphics window, the Command History window and the Data Sheet window. Across the top of the main windows there are menus for executing Dataplot commands. Across the bottom is a command entry window where commands can be typed in.
Data Analysis Steps Results and Conclusions

Click on the links below to start Dataplot and run this case study yourself. Each step may use results from previous steps, so please be patient. Wait until the software verifies that the current step is complete before clicking on the next step.

The links in this column will connect you with more detailed information about each analysis step from the case study description.

```1. Get set up and started.
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```   1. Read in the data.

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```
```
```
```
``` 1. You have read 2 columns of numbers
into Dataplot, variables the
ultrasonic response and metal
distance
```
```2. Plot data, pre-fit for starting values, and
fit nonlinear model.
```
```   1. Plot the ultrasonic response versus
metal distance.

```
```   2. Run PREFIT to generate good
starting values.

```
```   3. Nonlinear fit of the ultrasonic response
versus metal distance.  Plot predicted
values and overlay the data.
```
```   4. Generate a 6-plot for model
validation.

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```   5. Plot the residuals against
the predictor variable.

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```

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``` 1. Initial plot indicates that a
nonlinear model is required.
Theory dictates an exponential
over linear for the initial model.

```
``` 2. Pre-fit indicated starting
values of 0.1 for all 3
parameters.

```
``` 3. The nonlinear fit was carried out.
Initial fit looks pretty good.

```
``` 4. The 6-plot shows that the model
assumptions are satisfied except for
the non-homogeneous variances.
```
``` 5. The detailed residual plot shows
the non-homogeneous variances
more clearly.
```
```3. Improve the fit with transformations.
```
```   1. Plot several common transformations
of the dependent variable (ultrasonic
response).

```
```   2. Plot several common transformations
of the predictor variable (metal).

```
```   3. Nonlinear fit of transformed data.
Plot predicted values with the
data.

```
```   4. Generate a 6-plot for model
validation.

```
```   5. Plot the residuals against
the predictor variable.

```
```
```
``` 1. The plots indicate that a square
root transformation on the dependent
variable (ultrasonic response) is a
good candidate model.
```
``` 2. The plots indicate that no
transformation on the predictor
variable (metal distance) is
a good candidate model.
```
``` 3. Carry out the fit on the transformed
data.  The plot of the predicted
values overlaid with the data
indicates a good fit.

```
``` 4. The 6-plot suggests that the model
assumptions, specifically homogeneous
variances for the errors, are
satisfied.

```
``` 5. The detailed residual plot shows
more clearly that the homogeneous
variances assumption is now
satisfied.
```
```4. Improve the fit using weighting.
```
```   1. Fit function to determine appropriate
weight function.  Determine value for
the exponent in the power model.

```
```   2. Plot residuals from fit to determine
appropriate weight function.

```
```   3. Weighted linear fit of field versus
lab.  Plot predicted values with
the data.

```
```   4. Generate a 6-plot for model
validation.

```
```   5. Plot the residuals against
the predictor variable.

```
```
```
``` 1. The fit to determine an appropriate
weight function indicates that a
value for the exponent in the range
-1.0 to -1.1 should be reasonable.
```
``` 2. The residuals from this fit
indicate no major problems.

```
``` 3. The weighted fit was carried out.
The plot of the predicted values
overlaid with the data suggests
that the variances arehomogeneous.
```
``` 4. The 6-plot shows that the model
assumptions are satisfied.

```
``` 5. The detailed residual plot suggests
the homogeneous variances for the
errors more clearly.
```
```5. Compare the fits.
```
```   1. Plot predicted values from each
of the three models with the
data.

```
```
```
``` 1. The transformed and weighted fits
generate only slightly different
predicted values, but the model
assumptions are not violated.

```